Implicit Hybrid Block Collocation Method for the Solution of Volterra Integral Equation of the Second Kind

Full Article - PDF

Published: 2023-07-05

Page: 218-228


D. Raymond

Department of Mathematics and Statistics, Federal University Wukari, Nigeria.

A. Adu *

Department of Mathematics and Statistics, Federal University Wukari, Nigeria.

R. Ajia

Department of Mathematics and Statistics, College of Agriculture, Science and Technology, Jalingo, Nigeria.

*Author to whom correspondence should be addressed.


Abstract

This paper proposes a three-step hybrid block method with two off-grid collocation points that solves Volterra integral equation of the second kind. The collocation of power series and trigonometric approach was used as the basis function to generate the continuous hybrid linear multistep method, which was then evaluated at non-interpolating points to give a continuous block method. The discrete block method was recovered when the continuous block was evaluated at all step points. The basic properties of the methods were investigated and were found to be zero-stable, consistent and convergent. The effectiveness of the developed three-step method is applied to solve some Volterra integral equations of the second kind by converting into an initial value problems of ordinary differential equations and from the numerical results obtained, it is observed that our methods gives better approximation than the existing method compared with.

Keywords: Trigonometric, hybrid point, second kind, Volterra integral, power series


How to Cite

Raymond , D., Adu , A., & Ajia , R. (2023). Implicit Hybrid Block Collocation Method for the Solution of Volterra Integral Equation of the Second Kind. Asian Journal of Pure and Applied Mathematics, 5(1), 218–228. Retrieved from https://globalpresshub.com/index.php/AJPAM/article/view/1826

Downloads

Download data is not yet available.

References

Gonzalez-Rodelas P. Pasadas M, Kouibia A, Mustafa B. Numerical solution of linear volterra integral equation system of second kind by Radial Basis function. Mathematics. 2022;10(2):223.

Wazwaz AM. Linear and Nonlinear Integral Equations: Methods and Applications, Springer; 2011.

Biazar J, Eslami M. Modified HPM for solving systems of Volterra integral equations of the second kind, J. King Saud Univ. Sci. 2011;23(1):35–39.

Tahmasbi A, Fard OS. Numerical solution of linear Volterra integral equations system of the second kind, Appl. Math. Comput. 2008;201(1):547–552.

Aminikhan H, Biazar J. A new analytical method for solving systems of Volterra integral equations. International Journal of Compute. Mathematics. 2010;87(5)1142–1157.

Bhattacharya S, Mandal BN. Use of bernstein polynomials is numerical solution of volterra integral equations. Applied Mathematical Sciences. 2008;2(36):1773-1787.

Berenguer MI, Gámez D, Garralda-Guillém AI, Ruiz Galán M, Serrano Pérez MC. Analytical Techniques for a Numerical Solution of the Linear Volterra Integral Equation of the Second Kind. In Abstract and Applied Analysis. Hindawi; 2009.

Mirzaae F. Numerical computational solution of the linear Volterra integral equations systems via rationalized Haar functions. Journal of King Saud University. 2010;22(4):265-268.

Sahn, N, Yüzba S, Glsu MA. Collocation approach for solving systems of linear Volterra integral equations with variable coefficients. Comput. Math. Appl. 2011;62:755–769.

Yang LH, Shen H, Wang Y. The reproducing kernel method for solving the system of the linear Volterra integral equations with variable coefficients. J. Compute. Appl. Math. 2012;236:2398–2405.

Maturi DA, Bajamal AZ, Algethami B. Numerical solution of volterra Integral Equation of the second kind using implicit trapezoidal. Journal: Journal of Advances In mathematics. 2014;8(2).

Balakumar V, Murugesan K. Numerical solution of volterra integral-algebraic equations using block pulse functions. Applied Mathematics and Computer. 2015;263:165–170.

Islam MS, Bangalee MZI, Khan AK, Halder A. Approximate Solution of systems of Volterra Integral Equations of Second Kind by Adomian Decomposition Method. Dhaka. Univ. J. Sci. 2015;63:15–18.

Shoukralla ES, Ahmed BM. Numerical solutions of volterra integral equations of the second using Lagrange interpolation via thee vandermonde matrix. In Journal of Physics: Conference Series. IOP Publish. 2020;1447(1): 12003.

Rouibah K, Bellour A, lima P, Rawashden E. Iterative continuous collocation for solving nonlinear volterra integral equations. Kragujevac Journal of Mathematics. 2022;46(4):635-645.